Circle Geometry: Tangents, Chords and Secants

In short: A tangent touches a circle at exactly one point and is perpendicular to the radius there. Two tangents from the same outside point are equal in length, and the alternate segment theorem says the angle between a tangent and a chord equals the angle in the alternate segment.

Starting point: Years 9 and 10

Tangent words

A tangent touches a circle at exactly one point, the point of contact.
A secant is a line that cuts the circle at two points.

Tangent theorems

PA = PB, and each tangent meets its radius at 90°.

Intersecting chords, secants and tangents

Tangents and secants

Tangent

Touches once
Perpendicular to radius
Two tangents equal

Alternate segment

Tangent-chord angle
Equals angle opposite

Products

Chords: AP × PB = CP × PD
Secants: PA × PB = PC × PD
Tangent: PT² = PA × PB

Quick summary

A tangent touches the circle once and is perpendicular to the radius there.
Two tangents from one outside point are equal.
Alternate segment theorem: tangent-chord angle equals the angle in the alternate segment.
Chords: $A P \times P B = C P \times P D$ . Secants: $P A \times P B = P C \times P D$ . Tangent: $P T^{2} = P A \times P B$ .

Quiz

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Try 3 questions now

A tangent meets the radius at the point of contact at:
1. A $4 5^{\circ}$
2. B $9 0^{\circ}$
3. C $18 0^{\circ}$
4. D any angle
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B: $9 0^{\circ}$
Tangent and radius are perpendicular.
A line that cuts a circle at two points is a:
1. A tangent
2. B secant
3. C radius
4. D diameter only
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B: secant
A secant passes through the circle.
Radius 6, and $P$ is 10 from the centre. The tangent length from $P$ is:
1. A 4
2. B 8
3. C 16
4. D $136$
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B: 8
$100 - 36 = 8$ .